DIMENSIONSFrom left to right, the square, the cube, and the
tesseract. The square is bounded by 1-dimensional lines, the cube
by 2-dimensional areas, and the tesseract by 3-dimensional volumes.
A proection of the cube is given since it is vie!ed on a
t!o-dimensional screen. The same applies to the tesseract, !hich
additionally can only be sho!n as a proection even in
three-dimensional space.A diagram sho!ing the first four spatial
dimensions."n mathematics and physics, the dimension of a space or
obect is informally defined as the minimum number of coordinates
needed to specify each point !ithin it.#1$#2$ Thus a line has a
dimension of one because only one coordinate is needed to specify a
point on it. A surface such as a plane or the surface of a cylinder
or sphere has a dimension of t!o because t!o coordinates are needed
to specify a point on it %for e&le, to locate a point on the
surface of a sphere you need both its latitude and its longitude'.
The inside of a cube, a cylinder or a sphere is three-dimensional
because three co-ordinates are needed to locate a point !ithin
these spaces."n physical terms, dimension refers to the constituent
structure of all space %cf. volume' and its position in time
%perceived as a scalar dimension along the t-a&is', as !ell as
the spatial constitution of obects !ithin (structures that have
correlations !ith both particle and field conceptions, interact
according to relative properties of mass, and !hich are
fundamentally mathematical in description. These or other a&es
may be referenced to uniquely identify a point or structure in its
attitude and relationship to other obects and events. )hysical
theories that incorporate time, such as general relativity, are
said to !or* in +-dimensional ,spacetime,, %defined as a -in*o!s*i
space'. -odern theories tend to be ,higher-dimensional, including
quantum field and string theories. The state-space of quantum
mechanics is an infinite-dimensional function space. The concept of
dimension is not restricted to physical obects. .igh-dimensional
spaces occur in mathematics and the sciences for many reasons,
frequently as configuration spaces such as in /agrangian or
.amiltonian mechanics0 these are abstract spaces, independent of
the physical space !e live in.Contents 1"n mathematics o
1.11imension of a vector spaceo 1.2-anifoldso 1.3/ebesgue covering
dimensiono 1.+"nductive dimensiono 1.2.ausdorff dimensiono
1.3.ilbert spaces 2"n physics o 2.14patial dimensionso 2.2Timeo
2.3Additional dimensions 3/iterature +)hilosophy 2-ore dimensions
34ee also o 3.1A list of topics inde&ed by dimension
56eferences 7Further readingIn mathematics"n mathematics, the
dimension of an obect is an intrinsic property, independent of the
space in !hich the obect may happen to be embedded. For
e&le8 a point on the unit circle in the plane can be
specified by t!o 9artesian coordinates but one can ma*e do !ith a
single coordinate %the polar coordinate angle', so the circle is
1-dimensional even though it e&ists in the2-dimensional plane.
This intrinsic notion of dimension is one of the chief !ays in
!hich the mathematical notion of dimension differs from its common
usages.The dimension of :uclidean n -space E n is n. ;hen trying to
generali, ?ne ans!er is that to cover a fi&ed ball in E n by
small balls of radius , one needs on the order of @n such small
balls. This observation leads to the definition of the -in*o!s*i
dimension and its more sophisticated variant, the .ausdorff
dimension. Aut there are also other ans!ers to that question. For
e&le, one may observe that the boundary of a ball in E n
loo*s locally li*e E n @ 1 and this leads to the notion of the
inductive dimension. ;hile these notions agree on E n, they turn
out to be different !hen one loo*s at more general spaces.A
tesseract is an e&le of a four-dimensional obect. ;hereas
outside of mathematics the use of the term ,dimension, is as in8 ,A
tesseract has four dimensions,, mathematicians usually e&press
this as8 ,The tesseract has dimension 4,, or8 ,The dimension of the
tesseract is +.,Although the notion of higher dimensions goes bac*
to 6enB 1escartes, substantial development of a higher-dimensional
geometry only began in the 1Cth century, via the !or* of Arthur
9ayley,;illiam 6o!an .amilton, /ud!ig 4chlDfli and Aernhard
6iemann. 6iemannEs 172+ .abilitationsschrift, 4chlafiEs 1722
Theorie der vielfachen Kontinuitt, .amiltonEs 17+3 discovery of the
quaternions and the construction of the 9ayley Algebra mar*ed the
beginning ofhigher-dimensional geometry.The rest of this section
e&amines some of the more important mathematical definitions of
the dimensions.Dimension of a vector spaceMain article: Dimension
(vector space)The dimension of a vector space is the number of
vectors in any basis for the space, i.e. the number of coordinates
necessary to specify any vector. This notion of dimension %the
cardinality of a basis' is often referred to as the Hamel dimension
or ale!raic dimension to distinguish it from other notions of
dimension.ManifoldsA connected topological manifold is locally
homeomorphic to :uclidean n-space, and the number n is called the
manifoldEs dimension. ?ne can sho! that this yields a uniquely
defined dimension for every connected topological manifold.The
theory of manifolds, in the field of geometric topology, is
characteri
